Proceedings of the ASME 2016 Conferences on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2016 September 28-30, 2016, Stowe, VT, USA
SMASIS2016-9137
OPERATIONAL SPACE CONTROL OF A LIGHTWEIGHT ROBOTIC ARM ACTUATED BY SHAPE MEMORY ALLOY (SMA) WIRES
Serket Quintanar-Guzm ´an∗
Interdisciplinary Centre for Security, Reliability and Trust University of Luxembourg
Luxembourg, 1359 Luxembourg
Email: serket.quintanar@uni.lu
Somasundar Kannan, Miguel A. Olivares-Mendez,
Holger Voos
Interdisciplinary Centre for Security, Reliability and Trust University of Luxembourg
Luxembourg, 1359 Luxembourg
ABSTRACT
This paper presents the design and control of a two link lightweight robotic arm using a couple of antagonistic Shape Memory Alloy (SMA) wires as actuators. A nonlinear robust con- trol law for accurate positioning of the end effector of the two- link SMA based robotic arm is developed to handle the hysteresis behavior present in the system. The model presented consists of two subsystems: firstly the SMA wires model and secondly the dynamics of the robotic arm itself. The control objective is to po- sition the robotic arm’s end effector in a given operational plane position. For this regulation problem a sliding mode control law is applied to the hysteretic system. Finally a Lyapunov analy- sis is applied to the closed-loop system demonstrating the stabil- ity of the system under given conditions. The simulation results demonstrate the accurate and fast response of the control law for position regulation. In addition, the stability of the closed-loop system can be corroborated.
INTRODUCTION
Shape Memory Alloys (SMA) consist of a group of metal- lic materials which can return to their initial shape and size after been deformed, when subjected to the appropriate thermal pro- cedure. This phenomenon is known as Shape Memory Effect
∗Address all correspondence to this author.
(SME). The Niquel-Titanium SMA wires (known as NiTi wires) are one of the most common types of SMA.
The SME occurs due to an inner transformation of the mate- rial’s crystalline structure. This transformation happens between two phases called martensite and austenite. When the SMA wire is at lower temperature its structure shifts to the martensite phase which is a relatively soft and malleable phase, during wich the wire can be easily deformed. When heated over the transfor- mation temperature the SMA wire transforms into the austenite phase, a hard phase, recovering its initial form and size.
The high force to mass ratio, small size, noiseless operation, and bio-compatibility have made NiTi wires a great alternative to conventional hydraulic, electric and pneumatic actuators. How- ever, they still represent a huge challenge for accurate implemen- tation, due to its slow and highly non-linear response.
Nowadays SMA wires are being used in a wide range of applications, from art-craft designs [1], to medical implements like intra-arterial supports [2] or dental applications [3], con- struction vibrations damper [4], micro-robotics [5, 6], specific purpose actuators as camera lense focus actuator [7], car mirror actuators [8] or SMA based motors [9], general purpose actua- tors [10–15] and robotic manipulators as arms, hands or robotic fingers [16–18]. Most of the mentioned applications are micro- scale or required complicated mechanical systems to be imple- mented. This paper presents a new design for a SMA wire ac-
tuated robotic arm. Our proposal keeps the simplicity of the mechanics and therefore achieves a light-weight actuator. This light-weight characteristic is critical in applications like robotic manipulator for unmanned aerial vehicles (UAVs), wich will be the main purpose of this arm. It is a great challenge to make an optimal use of available payload of an aerial vehicle such as a quadcopter. We propose a lightweight design, which enables the arm to be implemented without significantly decreasing the quadcopter’s available payload.
The paper is organized as follows: Section describes the me- chanical design and mathematical model of the proposed robotic arm. Section presents the non-linear applied control law as well as the operational space control law, followed by the stability analysis developed in section . The results are discussed in sec- tion . Finally the conclusion are presented in Section .
SMA ACTUATED ROBOTIC ARM
This section presents the mechanical design of the SMA ac- tuated robotic arm as well as the mathematical model of the over- all system. The mathematical model of the SMA wires, as well as its kinematic and dynamic relation with the mechanical model is explained in detail. Finally the modeling approach for the me- chanical system based on the CAD (Computer Aided Design) model is illustrated.
Mechanical Design
In this section we present the mechanical design of a light- weight SMA actuated robot arm. The optimal use of available payload of an aerial vehicle is critical for aerial manipulators de- sign, with this in mind we propose a single Degree of Freedom (DOF) actuator activated by a couple of antagonistic SMA wires.
Figure 1 shows a CAD model of the robot arm design. The de- sign is based on an existing joint proposed in [12]. This joint consists of two couplers joined by a torsion spring. Each SMA wire is attached in one end to its respective coupler allowing to control independently the angular position of each coupler. The SMA wire attached to coupler 2 allows to increase or decrease the stiffness of the joint as required by adjusting the torque ap- plied by the torsion spring, while SMA 1 affects directly the an- gular position of the end effector (θ1) by controlling the angular position of coupler 1 (see Fig. 1).
The given robotic arm with 1 DOF, is activated by two 37 cm long NiTi wires. It has two custom made carbon fiber links (150 mm and 100 mm respectively) and a range of movement along the vertical plane X -Z up to 85 degrees with two 7.5mm radius couplers. It has a total weight of 48g, which is about 25% of the weight respect to other light-weight designs found in the lit- erature as the one presented in [19]. The winding wheels enable the use of longer SMA wires to increase the movement range without increasing the dimension of the links.
FIGURE 1: PROPOSED SMA WIRE ACTUATED ROBOTIC ARM CAD MODEL
System Modeling
The overall system model is composed of several subsys- tems. Figure 2 shows the block diagram of the robotic arm’s mathematical model. This model is divided in two main subsys- tems: 1) Operational Space Control and 2) Actuator and inner control. The latter, in turn, is formed by three other subsys- tems: Inner control law, SMA Model and Kinematic and Dy- namic model. As shown in Fig. 2, the Actuator and Inner control subsystem has several cross-coupling effects and is considered as a MIMO (multiple-input and multiple-output) system with two inputs (V1,V2) and two outputs (θ1, θ2).
The Actuator and Inner control contains the SMA wire sub- system, and its schematic model is illustrated in Fig.3. This SMA wire subsystem is described by a mathematical model of the NiTi wires which was proposed by the authors in [11]. This is also di- vided in three subsystems representing the thermal dynamics, the heat transformation and the constitutive model. In this figure the interaction between the variables of each subsystem of the SMA
FIGURE 2: BLOCK DIAGRAM OF THE SMA ACTUATED ROBOTIC ARM
FIGURE 3: SMA WIRE MATHEMATICAL MODEL BLOCK DIAGRAM
wire model is shown. On the other hand, the dynamics of the arm is directly derived from a Computer Aided Design (CAD) model. Each of the mentioned subsystems will be explained in more detail in the following subsections.
Heat Transfer Model. The heat transfer block consists of the electrical heating (Joule effect) and the natural convection model described by the following equation [11]:
mwcp
dT dt =V2
R − hAw(T − Tamb) (1) where V is the voltage, R is the electric resistance per unit length, cpis the specific heat, mwis the mass per unit length, Awis the wire surface area, Tambthe ambient temperature and T the SMA wire temperature. Here h is approximated by a second order polynomial of the temperature:
h= h0+ h2T2 (2)
SMA Wire Phase Transformation Model. As shown in Fig. 3, the block containing the phase transformation model computes the martensite fraction (ξ ). The phase transformation of the SMA wire depends directly on the direction of the time derivative of the temperature. Therefore, due to hysteresis behav- ior two equations are needed to fully describe the phase transfor- mation phenomenon. The phase transformation from martensite to austenite while heating is given by
ξ =ξM
2 {cos [aA(T − As) + bAσ ] + 1} (3) for As+Cσ
A ≤ T ≤ Af+Cσ
Inversely, the transformation from austenite to martensite, duringA
cooling is described by the following dynamic equation:
ξ =1 − ξA
2 cos [aM(T − MF) + bMσ ] +1 + ξA
2 (4)
for Ms+Cσ
M ≤ T ≤ Mf+Cσ
where Ms, Mf, As, Af are the start and end transformationM
temperatures for martensite and austenite transformation respec- tively. And aA= π
(Af−As), aM= π
(Ms−Mf), bA= −CaA
A, bM= −CaM
M, CAand CMare curve fitting parameters.
Wire Constitutive Model. The wire constitutive model describes the relation between stress σ , strain ε, temperature T and martensite fraction ξ . The general form, firstly proposed in [20] and then modified in [11], is written as
σ = E ˙˙ ε + Ω ˙ξ + Θ ˙T (5)
The authors in [11] propose a constant value for the Young’s modulus E as an average of the Young’s modulus of each phase austenite (EA) and martensite (EM). However, since the actuator presented here uses antagonistic SMA wires, the Young’s modu- lus depends of the martensite fraction as follows:
E= ξ EM+ (1 − ξ ) EA (6)
here Ω and Θ represent the phase transformation constant and thermal expansion coefficient, respectively. Where
Ω = −Eε0 (7)
and ε0is the initial strain.
Kinematic Model and Dynamics. In this section the model of the mechanical design and its relation with the rest of the system is explained.
Kinematic Model. The kinematic model relates the SMA wire model with the mechanics of the robotic arm itself. The strain ratio of the SMA wire and angular velocity of the arm are related kinematically as:
ε˙i= −riθ˙i
l0i (8)
where ri is the respective coupler radius, l0the initial length of each wire and ˙θi the angular velocity of each coupler. Equation (8) shows that the angular position of each coupler with respect to the X -axis (θi) is inversely proportional to the strain of the wire (εi).
Dynamics. The dynamics describe the relation between coupler mechanism, torsion spring and forces applied by the SMA wires, as well as the effects of the load and grip at the end of the second link. The general dynamic model of the mechanical system is mathematically described as:
J ¨θ =
τw1(σ ) − τs(θ ) − τg(θ ) − τload(θ ) − b1θ˙1
−τw2(σ ) + τs(θ ) − b2θ˙2
(9)
where b is the friction of each coupler and τ is the torque ap- plied over the mechanical system by the SMA wires (w), the tor- sion spring (s), the weight of the gripper (g) and the load. This model was developed in two parts: Firstly the coupler mecha- nism, gripper, load and link dynamics and secondly the mathe- matical model of τw1and τs. The dynamic behavior of the cou- plers, gripper, load and links was directly obtained from the CAD design shown in Fig. 1 developed in Autodesk/Inventor environ- ment.
This model does not only include the exact geometry of each piece but masses, inertias and centers of mass necessary for dy- namic analysis. The CAD model is imported via the SimMe- chanics toolbox in order to obtain a continuous dynamic MAT- LAB/Simulink model of the mechanical system. On the other hand, the torsions spring and SMA wires torques were obtained from basic physical laws, from which it can be deduced that the SMA wire’s force (Fw) is inversely proportional to the stress (σ ) which can be calculated by integration of Eq. (5):
τwi= Fwiri= Aσiri (10)
where r is the coupler radius and A the transversal area of the
wire. The torsion spring torque τsis calculated as:
τs= ks(θ1− θ2) + bs θ˙1− ˙θ2
(11)
where ksis the spring constant and bsis the spring’s friction fac- tor, θ1and θ2are the angular position of each coupler with re- spect to X -axis.
CONTROL LAW
The control law presented here is divided into two separate controllers. First is the Inner control law, which works in the Joint Space, which regulates the output angle of the actuator θ1. Second there is the Operational Space Control, which allows to control the end effector poition in Cartesian coordinate system.
These two control laws are explained in further detail in the fol- lowing subsections.
Inner Control Law
The inner control law regulates the rotational movement of the end effector. The controlled variable is the angular position of coupler 1 (θ1) (see Fig. 1). For the inner control a Variable Struc- ture Control (VSC) is applied, which utilizes a switching control law to force the plant’s state trajectory onto a user-specified sur- face in the state space and maintain it on that surface. Depending if the state is over or under the surface, different control struc- tures are applied. This type of control is considered to be robust against parameter uncertainties, model nonlinearities and exter- nal disturbances.
The basic control law is given by
si= ci+ cpie+ cIi Z
edt (12)
where cpi and cIi represent the proportional and integral gains respectively, ci is a small voltage to keep the wires in tension when the position error is zero. The angular position error e is defined as:
e= θ1− θr (13)
The control voltage is defined as
vi=
ViH, si≥ φi
si, 0 ≤ si< φi 0, si< 0
(14)
where φiis the boundary layer and ViH is the maximum voltage.
FIGURE 4: OPERATIONAL SPACE CONTROL BLOCK DIA- GRAM
Operational Space Control
A second controller is designed in order to work in an Op- erational Space, as shown in Fig. 2. This second control has as input the references in a Cartesian coordinate system and gives as output the reference in joint space for the inner control. The schematic diagram of the operational space control is illustrated in Figure 4. This is described by the following equation:
˙
q= JAT(q) Kreos (15)
where xdis the set of Cartesian coordinates for the end effector’s desired position, JAis the analytical Jacobian of the robotic arm, Kr∈ Rnsuch is a symmetric gain matrix and eosis the operational space error defined as
eos= xd− xe (16)
This control represents a simple proportional control which takes into account the direct and inverse dynamics of the one DOF robot arm. Equation (17) shows the analytical Jacobian of the robotic system and Eqn. (18) shows the direct kinematics.
JA(q) =∂ K (·)
∂ q = −a1sin (q1)
−a1cos (q1)
(17)
K(·) =
a1cos (q1) a1sin (q1) − h
(18)
STABILITY ANALYSIS
Consider the nonlinear system as follows:
˙ x=
V 21
R−h1Aw(x1−Tamb) mwcp V 22
R−h2Aw(x2−Tamb) mwcp
x5 x6
F1r1−b1x5−τr−mloadgrloadcos(x3) J1
τr−F2r2−b2x6
J2
(19)
Where:
x1- Temperature of SMA wire 1 (T1) x2- Temperature of SMA wire 2 (T2) x3- Angular position of coupler 1 (θ1) x4- Angular position of coupler 2 (θ2) x5- Angular velocity of coupler 1 θ˙1 x6- Angular velocity of coupler 2 θ˙2
Applying a change of variable, the regulation error (position error defined previously in Eqn. (13)) is rewritten as:
e= x3− xd (20)
Let ¯x= x1x2e x4x5x6T
be the new state vector.
For the system derived from Eqn.(19) and Eqn.(20), a Lyapunov function candidate is proposed as follow:
V( ¯x) = P1x1+ P2x2+ P3e2+ P4x24+ P5(x5+ x6)2 (21) where Pi∈ R > 0, i = 1, 2, 3, 4, 5. Assuming Tamb> 0 for xi≥ Tambwhere i = 1, 2 then V ( ¯x) can be considered to be positive- definitive.
If ¯x(0) = Tamb Tamb0 0 0 0 then
V( ¯x(0)) = Tamb(P1+ P2) = xe (22) where xeis the equilibrium point at t = 0.
The time derivative of the candidate Lyapunov function is:
V˙( ¯x) = P1x˙1+P2x˙2+P3ex˙3+P4x4x˙5+P5[x5x˙5+ ˙x5x6+ x5x˙6+ x6x˙6] (23) With this result, the function candidate derivative is proven to be negative definite if the conditions set shown in Eqn.(24), Eqn.(25) and Eqn.(26) are fulfilled.
P1≥P5(x5max+ x6max) F1maxr1mwCp
J1Awh1(x1max− Tamb) (24)
FIGURE 5: OPEN-LOOP SYSTEM HYSTERESIS CURVE STRAIN - TEMPERATURE
P2≥P5(x5max+ x6max) F2maxr2mwCp
J2Awh2(x2max− Tamb) (25)
P4≤P5b(x5min+ x6min)2+ P5(x5min+ x6min) k1
J1x4minx6min − P1VH12 + P2VH22 mwCpRx4minx6min
(26) where ximin≤ xi≤ ximax. These conditions are achievable as long as the following condition is fulfilled
ViH<
s
P5(x5min+ x6min) [b (x5min+ x6min) + mloadgrload] (mwcpR) 2J1P1
(27) The closed-loop system of the plant and inner control is then proven to be asymptotically stable [21]. In our case, for the op- erational space control system, it is proved that if Kris a positive definite matrix the system is asymptotically stable [22]. Which concludes the proof that overall system is asymptotically stable.
SIMULATION RESULTS
The SMA actuated robotic arm open and closed-loop per- formance process was evaluated through simulations using Simulink/MATLAB.
The nonlinear hysteretic characteristics of the system is first evaluated by open loop simulation. This test was performed by applying to the SMA 1 the maximum voltage (V1H) while the SMA 2 has no voltage applied and then vice versa. The results of this simulation are shown in Fig. 5. The hysteresis loop gener- ated by this test is the major hysteresis loop. Which is considered a double loop hysteresis and it is generated by the antagonistic SMA wires.
In order to carry out a closed-loop analysis a position regu- lation was performed. The step response was tested with a series
TABLE 1: PARAMETERS OF THE SMA WIRE AN THE COMPLIANT ACTUATOR [11, 12, 23, 24]
Parameter Value Parameter Value
EM 28 GPa CA 10 M pa/oK
EA 75 GPa CM 10 M pa/oK
As 88oC Tamb 25oC
Af 98oC A 4.9x10−8m2
Ms 72oC Aw 290.45x10−6m2
Mf 62oC cp 320 J/KgoC
mw 6.8x10−4kg/m εL 2.3%
R 20 Ω/m h0 20
l0 0.37 m h2 0.001
bs 0.5 b1, b2 0.1
ks 0.0018 Nm/1o Θ -0.055
of increased and decreased steps every 40s with an amplitude computed by Eqn. (28) as shown in Fig. 6 (solid line)
xd=
a1cos (N (i) π/180) a1sin (N (i) π/180) − h
(28)
where N = [10, 20, 30, 40, 50, 40, 30], a1 is the longitude of the second link (150 mm) and h is the longitude of the first link (100 mm) plus the base high (50 mm). The origin of the system is set at the center of the upper face of the robotic arm’s base. The parameters of the system used for simulation are listed in Tab. 1 and were taken from manufacturer in [23, 24] and [11, 12]. The Operational Space Control gain (Kr) was set at 300 in order to achieve a fast response through this controller. The inner control was set as follows: The boundary layers φ1= 10 degrees, φ2= 7 degrees, and limit voltage V1H= 6.5 V, V2H= 6.5 V. The results of this simulation are shown in Fig. 6.
In Fig. 6 it can be seen that there is an overshoot. This is attributed to the low friction factor of the system. The maximum overshoot is 13.2% for positive steps and 6.5% for negative steps in X -axis (see Fig. 6a), while for Z-axis (see Fig. 6b) is 21% and 12.72% respectively. In addition, the maximum steady state error in X -axis is 0.7% and 1.6% for Z-axis (see Fig. 7), with an av- erage settling time of 6 seconds. These maximum overshoot and maximum steady state error occur at the furthest position from the initial position, this condition is attributed to the transforma-
(a) STEP RESPONSE IN X AXIS
(b) STEP RESPONSE IN Z AXIS
FIGURE 6: STEP RESPONSE IN CARTESIAN COORDINATE FRAME
tion to operational space due to differences in reference frames in the mechanical CAD model. Figure 8 illustrates the inputs of the system during the closed-loop test. The inputs are given in Volts and they are limited to avoid thermal damage to the SMA wires, which can destroy its memory effect. Both SMA wires have the same high voltage limit, however the boundary layer for each wire is set at different levels as mentioned before. The higher limit for φ1is fixed in order to achieve a faster response from the control law. SMA 2 adjusts the stiffness of the joint. This means that the SMA 2 does not actuate directly over the end effector, thus its rate of response is not as critical as SMA 1.
FIGURE 7: STEP RESPONSE ERROR IN X AND Z
FIGURE 8: VOLTAGE INPUT
CONCLUSION
We have presented a SMA wire actuated light-weight robotic arm, which is intended to be an alternative for flying ma- nipulators. This arm is actuated by a couple of antagonistic SMA wires. It has a total weight of 48g and a range of movement up to 85 degrees on the X − Z plane. Operational Space Control was applied for position regulation, while a sliding mode control was applied as inner control.
Open and closed-loop test were developed. The closed-loop simulation results showed the capability of the variable structure control to deal with the double loop hysteresis generated by the antagonistic SMA wires. The maximum overshoot was observed to be 13.2% in X -axis and 21% for Z-axis, as well as, the max- imum steady state error was 0.0123 m for X -axis and 0.0057 m for Z-axis, which correspond to the 0.7% and 1.6% respectively.
The average settling time was 6 seconds.
Future work will be orientated to construct and to test exper- imentally the presented design. In addition, the proposed SMA based robotic arm will be attached to a small quadcopter, which is assembled as flying manipulator. Furthermore an ON/OFF control will be developed to open and close the gripper. The gripper will be actuated by a third SMA wire biased with a com- pression spring, avoiding the use of electric motor or any other kind of actuators besides SMA wires.
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